Invertible Square Matrices and Determinants - Problem 2

Norm Prokup

How do you tell if a matrix is invertible? For example is this matrix invertible? Well we have a result that says square matrix a is invertible, if and only if its determinant is not 0. So you have to take its determinant to tell.

Determinant of a is; 2 5 3, -15 10 18, 20 50 30. Now, the first thing I want to observe is that it's a 3 by 3 determinant. These are hard in general to evaluate. I want to use some real operations or column operations to simplify it. I'm noticing a common factor along the bottom row, so I'm going to pull out that factor of 10 out of the determinant. Remember that's one of the real operations you can do that. 2 5 3, -15 10 18 and I'll be left with 2 5 3.

In our previous example, we showed that whenever 2 rows or 2 columns are identical, the determinant is going to be 0. This is just going to be 10 times 0 which is 0. Thus the determinant of a is 0 and it's not invertible.

Now let's if we can generalize this a little bit. Whenever one row is a constant multiple of another row, in a matrix, its determinant will always be 0 and it will be invertible. The same goes for columns, if any column is a constant multiple of another column, the determinant is going to be 0 and the matrix is not invertible.